6-demicube

In geometry, a 6-demicube or demihexeract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

Demihexeract (6-demicube)
Petrie polygon projection
Type	Uniform 6-polytope
Family	demihypercube
Schläfli symbol	{3,33,1} = h{4,34} s{21,1,1,1,1}
Coxeter diagrams	= =
Coxeter symbol	131
5-faces	44	12 {31,2,1} 32 {34}
4-faces	252	60 {31,1,1} 192 {33}
Cells	640	160 {31,0,1} 480 {3,3}
Faces	640	{3}
Edges	240
Vertices	32
Vertex figure	Rectified 5-simplex
Symmetry group	D6, [33,1,1] = [1+,4,34] [25]+
Petrie polygon	decagon
Properties	convex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM₆ for a 6-dimensional half measure polytope.

Coxeter named this polytope as 1₃₁ from its Coxeter diagram, with a ring on one of the 1-length branches, . It can named similarly by a 3-dimensional exponential Schläfli symbol ${\displaystyle \left\{3{\begin{array}{l}3,3,3\\3\end{array}}\right\}}$ or {3,3^3,1}.

Cartesian coordinates

Cartesian coordinates for the vertices of a demihexeract centered at the origin are alternate halves of the hexeract:

(±1,±1,±1,±1,±1,±1)

with an odd number of plus signs.

As a configuration

This configuration matrix represents the 6-demicube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-demicube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^[1][2]

The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.^[3]

D6		k-face	fk	f0	f1	f2	f3		f4		f5		k-figure	notes
A4		( )	f0	32	15	60	20	60	15	30	6	6	r{3,3,3,3}	D6/A4 = 32*6!/5! = 32
A3A1A1		{ }	f1	2	240	8	4	12	6	8	4	2	{}x{3,3}	D6/A3A1A1 = 32*6!/4!/2/2 = 240
A3A2		{3}	f2	3	3	640	1	3	3	3	3	1	{3}v( )	D6/A3A2 = 32*6!/4!/3! = 640
A3A1		h{4,3}	f3	4	6	4	160	*	3	0	3	0	{3}	D6/A3A1 = 32*6!/4!/2 = 160
A3A2		{3,3}		4	6	4	*	480	1	2	2	1	{}v( )	D6/A3A2 = 32*6!/4!/3! = 480
D4A1		h{4,3,3}	f4	8	24	32	8	8	60	*	2	0	{ }	D6/D4A1 = 32*6!/8/4!/2 = 60
A4		{3,3,3}		5	10	10	0	5	*	192	1	1		D6/A4 = 32*6!/5! = 192
D5		h{4,3,3,3}	f5	16	80	160	40	80	10	16	12	*	( )	D6/D5 = 32*6!/16/5! = 12
A5		{3,3,3,3}		6	15	20	0	15	0	6	*	32		D6/A5 = 32*6!/6! = 32

Images

orthographic projections

Coxeter plane	B6
Graph
Dihedral symmetry	[12/2]
Coxeter plane	D6	D5
Graph
Dihedral symmetry	[10]	[8]
Coxeter plane	D4	D3
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

Related polytopes

There are 47 uniform polytopes with D₆ symmetry, 31 are shared by the B₆ symmetry, and 16 are unique:

D6 polytopes
h{4,34}	h2{4,34}	h3{4,34}	h4{4,34}	h5{4,34}	h2,3{4,34}	h2,4{4,34}	h2,5{4,34}
h3,4{4,34}	h3,5{4,34}	h4,5{4,34}	h2,3,4{4,34}	h2,3,5{4,34}	h2,4,5{4,34}	h3,4,5{4,34}	h2,3,4,5{4,34}

The 6-demicube, 1₃₁ is third in a dimensional series of uniform polytopes, expressed by Coxeter as k₃₁ series. The fifth figure is a Euclidean honeycomb, 3₃₁, and the final is a noncompact hyperbolic honeycomb, 4₃₁. Each progressive uniform polytope is constructed from the previous as its vertex figure.

k₃₁ dimensional figures

n	4	5	6	7	8	9
Coxeter group	A3A1	A5	D6	E7	${\displaystyle {\tilde {E}}_{7}}$ = E7+	${\displaystyle {\bar {T}}_{8}}$=E7++
Coxeter diagram
Symmetry	[3−1,3,1]	[30,3,1]	[31,3,1]	[32,3,1]	[33,3,1]	[34,3,1]
Order	48	720	23,040	2,903,040	∞
Graph					-	-
Name	−131	031	131	231	331	431

It is also the second in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1_3k series. The fourth figure is the Euclidean honeycomb 1₃₃ and the final is a noncompact hyperbolic honeycomb, 1₃₄.

1_3k dimensional figures

Space	Finite				Euclidean	Hyperbolic
n	4	5	6	7	8	9
Coxeter group	A3A1	A5	D6	E7	${\displaystyle {\tilde {E}}_{7}}$=E7+	${\displaystyle {\bar {T}}_{8}}$=E7++
Coxeter diagram
Symmetry	[3−1,3,1]	[30,3,1]	[31,3,1]	[32,3,1]	[[33,3,1]]	[34,3,1]
Order	48	720	23,040	2,903,040	∞
Graph					-	-
Name	13,-1	130	131	132	133	134

Skew icosahedron

Coxeter identified a subset of 12 vertices that form a regular skew icosahedron {3, 5} with the same symmetries as the icosahedron itself, but at different angles. He dubbed this the regular skew icosahedron.^[4][5]